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Chapter 14: Problem 29

One assumption of the ideal gas law is that the atoms or molecules themselves occupy a negligible volume. Verify that this assumption is reasonable by considering gaseous xenon (Xe). Xenon has an atomic radius of \(2.0 \times 10^{-10} \mathrm{m} .\) For STP conditions, calculate the percentage of the total volume occupied by the atoms.

Short Answer

Expert verified
Xenon atoms occupy a negligible percentage of the total volume at STP.

Step by step solution

01

Find the Volume of One Xenon Atom

The volume of a sphere is given by the formula \( V = \frac{4}{3} \pi r^3 \), where \( r \) is the radius. Substituting the radius of a xenon atom (\( 2.0 \times 10^{-10} \) m), the volume \( V \) is \( \frac{4}{3} \pi (2.0 \times 10^{-10})^3 \). Calculate this to find the volume of one xenon atom.
02

Calculate Molar Volume at STP

At STP (0掳C and 1 atm), 1 mole of an ideal gas occupies 22.4 liters or \( 22.4 \times 10^{-3} \) cubic meters. This is the total volume that 1 mole of xenon gas occupies under standard conditions.
03

Find the Number of Atoms in a Mole

Avogadro's number \( (6.022 \times 10^{23}) \) tells us the number of atoms in a mole of substance. Use this to find the total volume occupied by xenon atoms in a mole by multiplying Avogadro's number by the volume of one xenon atom.
04

Calculate the Total Volume Occupied by Atoms

Multiply the volume of one xenon atom by the total number of atoms in a mole (Avogadro's number) to obtain the total volume occupied by the xenon atoms in one mole of gas.
05

Compute the Percentage Occupied by Atoms

Divide the total volume occupied by the xenon atoms (from Step 4) by the molar volume (from Step 2) and multiply by 100 to find the percentage of the total volume occupied by the xenon atoms.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Xenon Gas
Xenon is a noble gas, known for its stability and lack of reactivity. It belongs to the group 18 elements in the periodic table and is often used in lighting such as in flash lamps and arc lamps. Xenon, being a heavy monatomic gas, is not prone to forming compounds easily, making it ideal for certain scientific applications.
  • Xenon is denser than air and colorless, making it invisible under normal conditions.
  • It is found in trace amounts in the Earth鈥檚 atmosphere, making it relatively rare compared to other gases like nitrogen or oxygen.
Understanding the properties of xenon helps in calculating how it behaves in different conditions, especially under the assumptions of the ideal gas law.
Delving into Atomic Radius
The atomic radius is a measure of the size of an atom, typically expressed in meters. For xenon, the atomic radius is approximately \(2.0 \times 10^{-10}\) meters. This value represents how far the outer electrons are from the nucleus, playing a crucial role in chemical reactions and physical states.
  • Atomic radius can vary slightly depending on the type of chemical bonds an atom is involved in.
  • It provides insight into how closely atoms can pack together in a given volume.
When we calculate the volume of a xenon atom using its atomic radius, we assume the atom is a sphere. The formula \( V = \frac{4}{3} \pi r^3 \) calculates the approximate volume, offering a glimpse into the spatial requirements of xenon atoms in a gas form.
Grasping Molar Volume
Molar volume is the volume occupied by one mole of a substance. For an ideal gas, under standard temperature and pressure (STP: 0掳C and 1 atm pressure), the molar volume is 22.4 liters or \(22.4 \times 10^{-3}\) cubic meters. This serves as a benchmark for comparing the behavior of gases.
  • Standard temperature and pressure conditions allow for consistency in calculations involving gases.
  • Molar volume helps determine how much space a certain amount of gas will fill, essential for storage and application purposes.
Under these conditions, we can compare the actual volume occupied by xenon atoms to the volume predicted by the ideal gas law, giving us a percentage of volume occupied.
Significance of Avogadro's Number
Avogadro's number, approximately \(6.022 \times 10^{23}\), represents the number of atoms, ions, or molecules in one mole of substance. It is a fundamental constant in chemistry, making it easier to handle vast numbers of particles due to their unimaginably small size.
  • Allows chemists to work with macroscopic amounts of substance conveniently.
  • Critical for converting between moles and the number of individual atoms or molecules.
By understanding Avogadro's number, we can compute the total volume that xenon atoms occupy, as it represents the multiplication of the volume of a single atom by this constant, giving a concrete scale to atomic size and gas volume.

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Most popular questions from this chapter

A gas fills the right portion of a horizontal cylinder whose radius is \(5.00 \mathrm{cm} .\) The initial pressure of the gas is \(1.01 \times 10^{5} \mathrm{Pa}\) A frictionless movable piston separates the gas from the left portion of the cylinder, which is evacuated and contains an ideal spring, as the drawing shows. The piston is initially held in place by a pin. The spring is initially unstrained, and the length of the gas-filled portion is \(20.0 \mathrm{cm} .\) When the pin is removed and the gas is allowed to expand, the length of the gas-filled chamber doubles. The initial and final temperatures are equal. Determine the spring constant of the spring.

Hemoglobin has a molecular mass of 64500 u. Find the mass (in \(\mathrm{kg}\) ) of one molecule of hemoglobin.

The volume of an ideal gas is held constant. Determine the ratio \(P_{2} / P_{1}\) of the final pressure to the initial pressure when the temperature of the gas rises (a) from 35.0 to \(70.0 \mathrm{K}\) and (b) from 35.0 to \(70.0^{\circ} \mathrm{C}\).

The active ingredient in the allergy medication Claritin contains carbon (C), hydrogen (H), chlorine (Cl), nitrogen (N), and oxygen (O). Its molecular formula is \(\mathrm{C}_{22} \mathrm{H}_{23} \mathrm{ClN}_{2} \mathrm{O}_{2} .\) The standard adult dosage utilizes \(1.572 \times 10^{19}\) molecules of this species. Determine the mass (in grams) of the active ingredient in the standard dosage.

When you push down on the handle of a bicycle pump, a piston in the pump cylinder compresses the air inside the cylinder. When the pressure in the cylinder is greater than the pressure inside the inner tube to which the pump is attached, air begins to flow from the pump to the inner tube. As a biker slowly begins to push down the handle of a bicycle pump, the pressure inside the cylinder is \(1.0 \times 10^{5} \mathrm{Pa}\), and the piston in the pump is \(0.55 \mathrm{m}\) above the bottom of the cylinder. The pressure inside the inner tube is \(2.4 \times 10^{5}\) Pa. How far down must the biker push the handle before air begins to flow from the pump to the inner tube? Ignore the air in the hose connecting the pump to the inner tube, and assume that the temperature of the air in the pump cylinder does not change.
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